One Oscillator Subject to Stochastic Forces

Let a limit cycle oscillator be exposed to some weak random forces which may depend on the state variable X. The governing equation is a nonlinear stochastic equation  

Interpreting the random forces as a perturbation ep in (5.2.1), one may immediately write down the equation for 0, to the lowest-order approximation, as 

Thus gr(0, t) is a r-periodic function of 0 but stochastically dependent on t unlike the corresponding quantity Q (p,t) in (5.2.2). In the usual stochastic models. 

It is a well known fact (Stratonovich, 1967) that an equation of the form (5.5.2) is then equivalent to the Fokker-Planck equation 

Rigorously speaking, the above e should be read but we retain the above form and simply interpret e as some small parameter. 

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